Mr. Obama seems to have anticipated Mr. Brady. During his State of the Union message Tuesday night, the president briefly outlined his proposals before passing immigration reform over to Congress. His approach to the issue, which has vexed several previous presidents and Congresses, was notably less strident than his calls for action on a range of other issues, from gun control to cybersecurity.

Obama's careful handling of immigration reform reflects a do-no-harm strategy that seems to recognize deep suspicions about his motives among many GOP lawmakers, such as Brady.

“I think the president will invariably make it harder for both sides to come together because he sort of likes baiting Republicans on this,” Brady said after the speech. “If he wanted to see true immigration reform, he’d lock himself in the Oval Office until Congress got it done.”

“He is in a difficult spot,” says Rep. John Yarmouth (D) of Kentucky, who is part of a House group working on bipartisan immigration legislation. “It can’t be seen as an Obama plan, or Republicans won’t vote for it. I think he rightly said there are two bipartisan groups working on it, two bipartisan groups are going to advance plans and we should have a vote on that. And I think that’s the only way it can possibly become reality is that if it’s always perceived in both chambers as a bipartisan plan.”

But lying low cannot be easy for a president who promised immigration reform to the nation's Latino and Asian voters, who overwhelmingly backed Obama's reelection in November. Indeed, five undocumented immigrants were in the audience Tuesday night.

“It’s a Catch-22 for him,” Rep. Raul Labrador of Idaho, a leading Republican advocate in the House for immigration reform, said Tuesday night just outside the House chamber. “If he talks too much about it, then it looks like he’s pushing his agenda down our throat. And if he doesn’t talk enough, then he gets criticized for that.”

Many Republicans said Obama had handled the immigration issue deftly.

“I thought on comprehensive immigration reform, I thought his words were measured," Rep. Paul Ryan (R) of Wisconsin, House Budget Committee chairman and a longtime supporter of immigration issues, told CNN. "I think the tone and the words he took were productive on that front.”

Representative Labrador, meanwhile, went on to suggest a way forward on the contentious matter of whether undocumented immigrants can ever become US citizens.

While Obama argued for a path to citizenship for the unauthorized, Labrador recast the problem of citizenship in a way that might appeal to conservative lawmakers, saying what the US needs is not a separate citizenship path for illegal immigrants but rather a smarter legal immigration system that illegal immigrants could access.

“I am opposed to creating a new pathway to citizenship only for the illegal aliens," Labrador said. "If we can reform the immigration system so more people can actually immigrate to the United States, we can allow the people that are here illegally right now to actually take advantage of that existing pathway or whatever new pathways we create.”

Labrador’s ideas on immigration policy could reverberate among vocal conservatives in the House.

Rep. Thomas Massie (R) of Kentucky, who could be straight out of central casting as an immigration reform naysayer, says he is “very interested” in immigration and has learned much from Labrador.

Representative Massie says he’d like to see a functional agricultural worker program and would “like to take the folks that are in the shadows out of the shadows.” He opposes a pathway to citizenship just for illegal immigrants, but he doesn't object to undocumented immigrants obtaining citizenship through the legal immigration process.

Still, getting from goodwill to good legislation will be a long, long road. And Republicans worry that Obama may make it devilishly hard for them to get to yes.

Brady, the Texas congressman, said the president wasn’t “too bad” on antagonizing Republicans Tuesday night. But remember, he cautions, “we’ve got a few more days of his campaign yet. Don’t sell him short.”